Finite Group Databasebeta

← #119829

Subgroups — #119829

Order 512.

606 conjugacy classes of subgroups.

The subgroup lattice has 474 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21894C211yesmaximal, characteristic
2C2×#1561C211yesmaximal, characteristic
3C2×#1493C211yesmaximal, characteristic
4#10844C211yesmaximal, characteristic
5C2×#1309C211yesmaximal, characteristic
6#8041C211yesmaximal, characteristic
7#7897C211yesmaximal, characteristic
8C2×#533C2211yescharacteristic
9C2×#423C2211yescharacteristic
10C2×#423C2211yescharacteristic
11C2×#423C2211yescharacteristic
12C2×#423C2211yescharacteristic
13C22×C4×C8C2211yescharacteristic
14#2285C2211yescharacteristic
15#1988—21no—
16#1981—21no—
17#1894—22no—
18#1803—21no—
19#1803—21no—
20C2×#341—22no—
21#1561—22no—
22#1561C414yescomplemented
23C2×#332—21no—
24C2×#332—21no—
25#1493—22no—
26#1493—22no—
27C2×Q8⋊C8—22no—
28#1309—22no—
29#1296—21no—
30#1296—21no—
31#1279—22no—
32C22×M16—21no—
33C22×M16—21no—
34C22×M16—21no—
35C22×M16D412yes—
36C23×C8—21no—
37#533—21no—
38#533—21no—
39#533—21no—
40#533—21no—
41#533—21no—
42#533—21no—
43#533C2×C412yes—
44#533D412yes—
45C2×C4⋊Q8D411yescharacteristic
46C2×C4⋊Q8D411yescharacteristic
47C2×#125—21no—
48C2×C22⋊Q8—21no—
49C2×C22⋊Q8—21no—
50C2×C4×Q8—21no—
51C2×C4×Q8—21no—
52C2×#117—21no—
53C2×#117—21no—
54C22×C42C2311yescharacteristic
55#423—21no—
56#423—21no—
57#423—21no—
58#423—21no—
59#423—21no—
60#423—21no—
61#423—21no—
62#423—21no—
63#423—21no—
64#423—21no—
65#423—21no—
66#423—21no—
67#423—21no—
68#423C2×C412yes—
69#423C2×C412yes—
70#423D412yes—
71C2×C4⋊C8—21no—
72C2×C4⋊C8—21no—
73C2×C4⋊C8—21no—
74C2×C4⋊C8—21no—
75C2×C4⋊C8—21no—
76C2×C4⋊C8—21no—
77C2×C4⋊C8—21no—
78C2×C4⋊C8—21no—
79C2×#103—22no—
80C2×#103—22no—
81C2×C4×C8—21no—
82C2×C4×C8—21no—
83C2×C4×C8—21no—
84C2×C4×C8—21no—
85C2×C4×C8—21no—
86C2×C4×C8D412yes—
87#384D411yescharacteristic
88#384D411yescharacteristic
89#382—21no—
90#380—21no—
91#380—21no—
92C4×C4⋊C4—21no—
93C4×C4⋊C4—21no—
94#341—44no—
95#340—22no—
96#340—22no—
97#332—24no—
98#332—24no—
99Q8⋊C8—28no—
100C22×Q8—21no—
101C22×Q8—21no—
102C23×C4C2×D411yescharacteristic
103C23×C4#4111yescharacteristic
104C23×C4#4111yescharacteristic
105C2×M16—22no—
106C2×M16—22no—
107C2×M16—22no—
108C2×M16—22no—
109C2×M16—22no—
110C2×M16—41no—
111C2×M16—41no—
112C2×M16—41no—
113C2×M16—42no—
114C2×M16—42no—
115C2×M16—42no—
116C2×M16—42no—
117C2×M16—81no—
118C2×M16—81no—
119C2×M16—81no—
120C2×M16—81no—
121C22×C8—22no—
122C22×C8—22no—
123C22×C8—22no—
124C22×C8—22no—
125C22×C8—41no—
126C22×C8—41no—
127C22×C8—41no—
128C22×C8—41no—
129C22×C8—41no—
130C22×C8—41no—
131C22×C8—41no—
132C4⋊Q8—21no—
133C4⋊Q8—21no—
134C4⋊Q8—21no—
135C4⋊Q8—21no—
136C4⋊Q8—21no—
137C4⋊Q8—21no—
138C4⋊Q8C22⋊C412yes—
139C4⋊Q8C22⋊C412yes—
140#125—22no—
141#125—22no—
142#125—41no—
143#125—41no—
144C22⋊Q8—22no—
145C22⋊Q8—22no—
146C22⋊Q8—22no—
147C22⋊Q8—22no—
148C22⋊Q8—41no—
149C22⋊Q8—41no—
150C22⋊Q8—41no—
151C22⋊Q8—41no—
152C4×Q8—22no—
153C4×Q8—22no—
154C4×Q8—22no—
155C4×Q8—22no—
156C4×Q8—41no—
157C4×Q8—41no—
158C4×Q8—41no—
159C4×Q8—41no—
160#117—22no—
161#117—22no—
162#117—41no—
163#117—41no—
164#117—81no—
165#117—81no—
166C2×C4⋊C4—21no—
167C2×C4⋊C4—21no—
168C2×C4⋊C4—21no—
169C2×C4⋊C4—21no—
170C2×C4⋊C4—21no—
171C2×C4⋊C4—21no—
172C2×C4⋊C4—21no—
173C2×C4⋊C4—21no—
174C2×C4⋊C4—21no—
175C2×C4⋊C4—21no—
176C2×C4⋊C4—41no—
177C2×C4⋊C4—41no—
178C2×C22⋊C4—41no—
179C2×C22⋊C4—41no—
180C2×C42—41no—
181C2×C42—41no—
182C2×C42C22×C411yescharacteristic
183C2×C42C2×D411yescharacteristic
184C2×C42C2×D411yescharacteristic
185C2×C42C2×D411yescharacteristic
186C2×C42C2×D411yescharacteristic
187C2×C42C2×D411yescharacteristic
188C2×C42#4111yescharacteristic
189C2×C42#4111yescharacteristic
190C2×C42#4111yescharacteristic
191C2×C42#4111yescharacteristic
192C2×C42#4111yescharacteristic
193C2×C42#4111yescharacteristic
194C4⋊C8—22no—
195C4⋊C8—22no—
196C4⋊C8—22no—
197C4⋊C8—22no—
198C4⋊C8—22no—
199C4⋊C8—22no—
200C4⋊C8—41no—
201C4⋊C8—41no—
202C4⋊C8—41no—
203C4⋊C8—41no—
204C4⋊C8—41no—
205C4⋊C8—41no—
206C4⋊C8—41no—
207C4⋊C8—41no—
208C4⋊C8—41no—
209C4⋊C8—41no—
210#103—82no—
211#103—82no—
212C4×C8—22no—
213C4×C8—22no—
214C4×C8—41no—
215C4×C8—41no—
216#95—41no—
217#95—41no—
218C24#12411yescharacteristic
219C2×Q8—22no—
220C2×Q8—22no—
221C2×Q8—41no—
222C2×Q8—41no—
223C2×Q8—41no—
224C2×Q8—41no—
225C2×Q8—41no—
226C2×Q8—41no—
227C2×Q8—41no—
228C2×Q8—41no—
229C2×Q8—41no—
230C2×Q8—41no—
231C22×C4—21no—
232C22×C4—21no—
233C22×C4—21no—
234C22×C4—21no—
235C22×C4—21no—
236C22×C4—21no—
237C22×C4—21no—
238C22×C4—21no—
239C22×C4—21no—
240C22×C4—21no—
241C22×C4—21no—
242C22×C4—21no—
243C22×C4—41no—
244C22×C4—41no—
245C22×C4—41no—
246C22×C4—41no—
247C22×C4#11711yescharacteristic
248C22×C4C4×D411yescharacteristic
249C22×C4C4×D411yescharacteristic
250C22×C4#12011yescharacteristic
251C22×C4#12111yescharacteristic
252C22×C4#12111yescharacteristic
253C22×C4#12111yescharacteristic
254C22×C4#12111yescharacteristic
255C22×C4#12111yescharacteristic
256C22×C4#12311yescharacteristic
257C22×C4#12311yescharacteristic
258C22×C4#12311yescharacteristic
259C22×C4#12311yescharacteristic
260C22×C4#12311yescharacteristic
261C22×C4#12311yescharacteristic
262C22×C4#12611yescharacteristic
263C22×C4#12611yescharacteristic
264C22×C4#12611yescharacteristic
265M16—44no—
266M16—44no—
267M16—82no—
268M16—82no—
269M16—82no—
270M16—82no—
271M16—82no—
272M16—82no—
273C2×C8—24no—
274C2×C8—42no—
275C2×C8—42no—
276C2×C8—42no—
277C2×C8—42no—
278C2×C8—42no—
279C2×C8—42no—
280C2×C8—42no—
281C2×C8—42no—
282C2×C8—42no—
283C2×C8—42no—
284C2×C8—42no—
285C2×C8—42no—
286C2×C8—81no—
287C2×C8—81no—
288C2×C8—81no—
289C2×C8—81no—
290C2×C8—81no—
291C2×C8—81no—
292C4⋊C4—22no—
293C4⋊C4—22no—
294C4⋊C4—22no—
295C4⋊C4—22no—
296C4⋊C4—42no—
297C4⋊C4—42no—
298C4⋊C4—41no—
299C4⋊C4—42no—
300C4⋊C4—41no—
301C4⋊C4—41no—
302C4⋊C4—41no—
303C4⋊C4—42no—
304C4⋊C4—41no—
305C4⋊C4—41no—
306C4⋊C4—41no—
307C4⋊C4—41no—
308C4⋊C4—41no—
309C4⋊C4—41no—
310C4⋊C4—41no—
311C4⋊C4—41no—
312C4⋊C4—81no—
313C4⋊C4—81no—
314C22⋊C4—42no—
315C22⋊C4—42no—
316C22⋊C4—82no—
317C42—42no—
318C42—42no—
319C42—81no—
320C42—81no—
321C42C2×C22⋊C411yescharacteristic
322C42#11711yescharacteristic
323C42C4×D411yescharacteristic
324C42C4×D411yescharacteristic
325C42#12011yescharacteristic
326C42#12111yescharacteristic
327C42#12111yescharacteristic
328C42#12111yescharacteristic
329C42#12311yescharacteristic
330C42#12311yescharacteristic
331C42#12311yescharacteristic
332C42#12311yescharacteristic
333C42#12411yescharacteristic
334C42#12411yescharacteristic
335C42#12411yescharacteristic
336C42#12611yescharacteristic
337C23—21no—
338C23—21no—
339C23—21no—
340C23—21no—
341C23—21no—
342C23—21no—
343C23#38811yescharacteristic
344C23#39411yescharacteristic
345C23#39711yescharacteristic
346Q8—42no—
347Q8—42no—
348Q8—42no—
349Q8—42no—
350Q8—81no—
351Q8—81no—
352Q8—81no—
353Q8—81no—
354C2×C4—21no—
355C2×C4—21no—
356C2×C4—21no—
357C2×C4—21no—
358C2×C4—21no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—21no—
362C2×C4—21no—
363C2×C4—21no—
364C2×C4—21no—
365C2×C4—21no—
366C2×C4—21no—
367C2×C4—21no—
368C2×C4—21no—
369C2×C4—21no—
370C2×C4—21no—
371C2×C4—21no—
372C2×C4—21no—
373C2×C4—21no—
374C2×C4—21no—
375C2×C4—21no—
376C2×C4—21no—
377C2×C4—21no—
378C2×C4—21no—
379C2×C4—21no—
380C2×C4—21no—
381C2×C4—21no—
382C2×C4—21no—
383C2×C4—21no—
384C2×C4—42no—
385C2×C4—42no—
386C2×C4—81no—
387C2×C4—81no—
388C2×C4—81no—
389C2×C4—81no—
390C2×C4—81no—
391C2×C4—81no—
392C2×C4—81no—
393C2×C4—81no—
394C2×C4—81no—
395C2×C4—81no—
396C2×C4#38111yescharacteristic
397C2×C4#38611yescharacteristic
398C2×C4#38611yescharacteristic
399C2×C4#38811yescharacteristic
400C2×C4#38811yescharacteristic
401C2×C4#38811yescharacteristic
402C2×C4#39411yescharacteristic
403C2×C4#39411yescharacteristic
404C2×C4#39411yescharacteristic
405C2×C4#39711yescharacteristic
406C2×C4#39711yescharacteristic
407C2×C4#39711yescharacteristic
408C2×C4#45311yescharacteristic
409C2×C4#45411yescharacteristic
410C2×C4#45511yescharacteristic
411C2×C4#45611yescharacteristic
412C2×C4#47111yescharacteristic
413C2×C4#47111yescharacteristic
414C2×C4#47111yescharacteristic
415C2×C4#47111yescharacteristic
416C2×C4#47211yescharacteristic
417C2×C4#47311yescharacteristic
418C2×C4#47311yescharacteristic
419C2×C4#47311yescharacteristic
420C8—44no—
421C8—82no—
422C8—82no—
423C8—82no—
424C8—82no—
425C22—21no—
426C22—21no—
427C22—21no—
428C22—21no—
429C22—21no—
430C22—21no—
431C22—41no—
432C22—41no—
433C22—41no—
434C22—41no—
435C22#144011yescharacteristic
436C22#203911yescharacteristic
437C22#204011yescharacteristic
438C22#205111yescharacteristic
439C22#205211yescharacteristic
440C22#205211yescharacteristic
441C22#205211yescharacteristic
442C4—21no—
443C4—21no—
444C4—21no—
445C4—21no—
446C4—21no—
447C4—21no—
448C4—82no—
449C4—82no—
450C4—161no—
451C4—161no—
452C4#188211yescharacteristic
453C4#188311yescharacteristic
454C4#190711yescharacteristic
455C4#190811yescharacteristic
456C4#193011yescharacteristic
457C4#193011yescharacteristic
458C4#203711yescharacteristic
459C4#203811yescharacteristic
460C4#204611yescharacteristic
461C4#204711yescharacteristic
462C4#207811yescharacteristic
463C4#207911yescharacteristic
464C2—41no—
465C2—41no—
466C2#1053211yescharacteristic
467C2#1053311yescharacteristic
468C2#1053411yescharacteristic
469C2#1053511yescharacteristic
470C2#1116411yescharacteristic
471C2#1119211yescharacteristic
472C2#1119511yescharacteristic
subgroupG/H
center, socleC23#394
commutatorC2×C42C22×C4
Frattini, cosocleC22×C42C23
prime radicalC24#124
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C22×C42 ▷ C22 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
22. #1561C4semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#218942
C2×#15612
C2×#14932
#108442
C2×#13092
#80412
#78972